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Separation of points by classes of harmonic functions

Published online by Cambridge University Press:  24 October 2008

D. H. Armitage
Affiliation:
Department of Pure Mathematics, Queen's University, Belfast BT7 1NN, Northern Ireland
S. J. Gardiner*
Affiliation:
Department of Mathematics & Statistics, McGill University, Montreal, Quebec, Canada, H3A 2K6
I. Netuka
Affiliation:
Mathematical Institute, Charles University, Sokolovská 83, CS-186 00 Praha, Czech Republic
*
Permanent address: Department of Mathematics, University College, Dublin 4, Ireland

Abstract

We characterize those domains Ω in ℝN for which the positive harmonic functions on Ω separate the points of Ω.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Ancona, A.. Une propriété de la compactification de Martin d'un domaine Euclidien. Ann. Inst. Fourier (Grenoble) 29 (1979), 7190.CrossRefGoogle Scholar
[2]Bauer, H.. Aspects of linearity in the theory of function algebras. In Function Algebras, Proceedings of an International Symposium on Function Algebras, Tulane, 1965 (Scott, Foresman and Co., 1966), pp. 122137.Google Scholar
[3]Bear, H. S.. A geometric characterization of Gleason parts. Proc. Amer. Math. Soc. 16 (1965), 407412.CrossRefGoogle Scholar
[4]Bear, H. S.. Part metric and hyperbolic metric. Amer. Math. Monthly 98 (1991), 109123.CrossRefGoogle Scholar
[5]Benedicks, M.. Positive harmonic functions vanishing on the boundary of certain domains in Rn. Ark. Mat. 18 (1980), 5371.CrossRefGoogle Scholar
[6]Bliedtner, J. and Janssen, K.. Harnacksche Kegel und Metrik in harmonischen Räumen. Math. Ann. 198 (1972), 8597.CrossRefGoogle Scholar
[7]Doob, J. L.. Classical Potential Theory and its Probabilistic Counterpart (Springer-Verlag, 1984).CrossRefGoogle Scholar
[8]Eremenko, A. E. and Lyons, T. J.. Finely open sets in the limit set of a geometrically finite Kleinian group. In Approximation by Solutions of Partial Differential Equations (editors Fuglede, B. et al. ) (Kluwer, 1992), pp. 6167.CrossRefGoogle Scholar
[9]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[10]Hayman, W. K. and Kennedy, P. B.. Subharmonic Functions, vol. 1 (Academic Press, 1976).Google Scholar
[11]Helms, L. L.. Introduction to Potential Theory (Wiley, 1969).Google Scholar
[12]Herron, D. A.. The Harnack and other conformally invariant metrics. Kodai Math. J. 10 (1987), 919.CrossRefGoogle Scholar
[13]Herron, D. A. and Schiff, J. L.. Positive harmonic functions and complete metrics. Canad. Math. Bull. 32 (1989), 286297.CrossRefGoogle Scholar
[14]Köhn, J.. Die Harnacksehe Metrik in der Theorie der harmonischen Funktionen. Math. Z. 91 (1966), 5064.CrossRefGoogle Scholar
[15]Küran, Ü.. Study of superharmonic functions in Rn × (0, + ∞) by a passage to Rn+3. Proc. London Math. Soc. (3) 21 (1970), 614636.Google Scholar
[16]Leutwiler, H.. On a distance invariant under Möbius transformations in RN. Ann. A cad. Sci. Fenn. Ser. AlMath. 12 (1987), 317.Google Scholar
[17]Segawa, S.. Martin boundaries and Denjoy domains. Proc. Amer. Math. Soc. 103 (1988), 177183.CrossRefGoogle Scholar
[18]Stein, E. M. and Weiss, G.. Introduction to Fourier Analysis (Princeton University Press, 1971).Google Scholar
[19]Tanaka, H.. On Harnack's pseudo-distance. Hokkaido Math. J. 6 (1977), 302305.CrossRefGoogle Scholar